/pdf/handbook-of-magma-functions-5anhrdxxj3.pdf

Handbook of magma functions

We study a new monoid structure for Artin groups associated with finite Coxeter systems. Like the classical positive braid monoid, the new monoid is a Garside monoid. We give several equivalent constructions: algebraically, the new monoid arises when studying Coxeter systems in a “dual” way, replacing the pair ( W , S ) by ( W , T ), with T the set of all reflections; geometrically, it arises when looking at the reflection arrangement from a certain basepoint. In the type A case, we recover the monoid constructed by Birman, Ko and Lee.

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The dual braid monoid

A new presentation of the $n$-string braid group $B_n$ is studied. Using it, a new solution to the word problem in $B_n$ is obtained which retains most of the desirable features of the Garside-Thurston solution, and at the same time makes possible certain computational improvements. We also give a related solution to the conjugacy problem, but the improvements in its complexity are not clear at this writing.

A new approach to the word and conjugacy problems in the braid groups

We construct a new monoid structure for Artin groups associated with finite Coxeter systems. This monoid shares with the classical positive braid monoid a crucial algebraic property: it is a Garside monoid. The analogy with the classical construction indicates there is a ``dual'' way of studying Coxeter systems, where the pair (W,S) is replaced by (W,T), with T the set of all reflections. In the type A case, we recover the monoid constructed by Birman-Ko-Lee

Publisher Summary Crossings are suggested as they are in a picture of a highway overpass on a map The identity braid has a canonical representation in which two strands never cross Multiplication of braids is by juxtaposition, concatenation, isotopy, and rescaling This chapter discusses Artin's braid group, B n and its role in knot theory The chapter illustrates ways in which structure first discovered in the braid groups generalizes to structure in Garside groups, Artin groups and surface mapping class groups Artin's braid group is naturally isomorphic to the mapping class group of an n-times punctured disc The chapter also illustrates the topological concept of a braid and of a group of braids via the notion of a configuration space It then outlines the new developments in the area of B n mapping, thus illustrating how to pass from diffeomorphisms to geometric braids and back again

Braids: A Survey

The braid groups are infinite non-commutative groups naturally arising from geometric braids The aim of this article is twofold One is to show that the braid groups can serve as a good source to enrich cryptography The feature that makes the braid groups useful to cryptography includes the followings: (i) The word problem is solved via a fast algorithm which computes the canonical form which can be efficiently manipulated by computers (ii) The group operations can be performed efficiently (iii) The braid groups have many mathematically hard problems that can be utilized to design cryptographic primitives The other is to propose and implement a new key agreement scheme and public key cryptosystem based on these primitives in the braid groups The efficiency of our systems is demonstrated by their speed and information rate The security of our systems is based on topological, combinatorial and group-theoretical problems that are intractible according to our current mathematical knowledge The foundation of our systems is quite different from widely used cryptosystems based on number theory, but there are some similarities in design

/pdf/new-public-key-cryptosystem-using-braid-groups-4jt4e0sw77.pdf

New Public-Key Cryptosystem Using Braid Groups

We implement various computations in the braid groups via practically efficient and theoretically optimized algorithms whose pseudo-codes are provided. The performance of an actual implementation under various choices of parameters is listed.

/pdf/an-efficient-implementation-of-braid-groups-2l6j5lgdh0.pdf

An Efficient Implementation of Braid Groups

A large class of positive finite presentations of the braid groups is found and studied. It is shown that no presentations but known exceptions in this class have the property that equivalent braid words are also equivalent under positive relations.

Jae Woo Han

Papers

New Public-Key Cryptosystem Using Braid Groups

An Efficient Implementation of Braid Groups

Positive presentations of the braid groups and the embedding problem

Positive presentations of the braid groups and the embedding problem